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Abstract
Row echelon form (REF) and reduced row echelon form (RREF) are among the most fundamental tools in linear algebra. Although they are commonly introduced as methods for solving systems of linear equations, their influence extends far beyond elementary computations. In algebraic geometry, row reduction appears naturally in the study of affine and projective varieties, determinantal varieties, tangent spaces, Gr¨obner bases, and rank stratifications. In this paper, we explore how Gaussian elimination provides a geometric language for understanding these structures. We discuss the relationship between REF and linear morphisms, dimension theory, tangent spaces, and low-rank matrix varieties. We also explain how Gr¨obner basis theory can be viewed as a nonlinear analogue of row reduction. Throughout the paper, we emphasize intuition, geometric interpretation, and illustrative examples that connect elementary linear algebra with modern algebraic geometry.
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